An inverse backward problem for degenerate two-dimensional parabolic equation

Abstract

This paper deals with the determination of an initial condition in the degenerate two-dimensional parabolic equation $\partial_{t}u-\mathrm{div}\left(a(x,y)I_2\nabla u\right)=f,\quad (x,y)\in\Omega,\; t\in(0,T),$ where $\Omega$ is an open, bounded subset of $\mathbb{R}^2$, $a \in C^1(\bar{\Omega})$ with $a\geqslant 0$ everywhere, and $f\in L^{2}(\Omega \times (0,T))$, with initial and boundary conditions $u(x,y,0)=u_0(x,y), \quad u\mid_{\partial\Omega}=0,$ from final observations. This inverse problem is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. To show the convergence of the descent method, we prove the Lipschitz continuity of the gradient of the Tikhonov functional. Also we present some numerical experiments to show the performance and stability of the proposed approach.